Question

evaluate the expression using a calculator. round your answer to two decimal places when appropriate.
$left( sqrt5{-8}
ight)^8 approx square$

Explanation:

Step1: Simplify the fifth root

First, find the fifth root of -8. The fifth root of -8, denoted as $\sqrt[5]{-8}$, can be written as $(-8)^{\frac{1}{5}}$. Calculating this, we know that $(-2)^5 = -32$ is not -8, wait, actually $(-8)^{\frac{1}{5}}=-8^{\frac{1}{5}}\approx - 1.5157$ (using a calculator to find the fifth root of 8 and then applying the negative sign).

Step2: Raise to the 8th power

Now, raise this result to the 8th power: $(-1.5157)^{8}$. Since an even power of a negative number is positive, we calculate $1.5157^{8}$. Using a calculator, $1.5157^{8}\approx(1.5157^{2})^{4}\approx(2.2974)^{4}\approx(2.2974^{2})^{2}\approx(5.2781)^{2}\approx27.857$. Wait, alternatively, we can use exponent rules: $(\sqrt[5]{-8})^{8}=(-8)^{\frac{8}{5}}=(-8)^{1+\frac{3}{5}}=(-8)^{1}\times(-8)^{\frac{3}{5}}=-8\times(-8)^{\frac{3}{5}}$. But $(-8)^{\frac{3}{5}}=((-8)^{\frac{1}{5}})^{3}\approx(-1.5157)^{3}\approx - 3.476$, so $-8\times(-3.476)\approx27.81$. Wait, maybe a better way is to use the calculator directly: first compute $\sqrt[5]{-8}\approx - 1.515716567$, then raise to the 8th power: $(-1.515716567)^{8}\approx(1.515716567)^{8}$. Calculating $1.515716567^{2}=2.297448714$, $2.297448714^{2}=5.278099203$, $5.278099203^{2}\approx27.857$. Wait, but let's do it step by step with the calculator:

First, $\sqrt[5]{-8}=-8^{1/5}\approx - 1.515716567$.

Then, raise to the 8th power: $(-1.515716567)^{8}= (1.515716567)^{8}$.

Calculating $1.515716567^{8}$:

$1.515716567^{2}=2.297448714$

$1.515716567^{4}=(2.297448714)^{2}\approx5.278099203$

$1.515716567^{8}=(5.278099203)^{2}\approx27.857$. Wait, but let's check with the exponent rule: $(-8)^{\frac{8}{5}}=(-8)^{1.6}=e^{1.6\ln(8)}\times e^{1.6\ln(-1)}$? No, wait, complex numbers? Wait, no, in real numbers, the fifth root of a negative number is real (odd root), so $(-8)^{\frac{8}{5}}$ is real because the exponent is a rational number with odd denominator, so the principal root is real. So using a calculator, $\sqrt[5]{-8}\approx - 1.515716567$, then $(-1.515716567)^{8}\approx(1.515716567)^{8}\approx27.86$ (rounded to two decimal places). Wait, but let's compute it more accurately:

$1.515716567^{8}$:

First, $1.515716567^2 = 2.297448714$

$1.515716567^4=(2.297448714)^2 = 5.278099203$

$1.515716567^8=(5.278099203)^2 = 27.857\approx27.86$ (rounded to two decimal places). Wait, but maybe I made a mistake in the exponent. Wait, $(\sqrt[5]{-8})^8 = (-8)^{8/5}=(-8)^{1 + 3/5}=(-8)\times(-8)^{3/5}$. $(-8)^{3/5}=((-8)^{1/5})^3\approx(-1.5157)^3\approx - 3.476$, so $(-8)\times(-3.476)=27.808\approx27.81$. Wait, there is a discrepancy. Let's use the calculator directly:

Calculate $\sqrt[5]{-8}$: $\sqrt[5]{-8}=-8^{1/5}\approx - 1.515716567$

Then, $(-1.515716567)^8$: since 8 is even, it's the same as $(1.515716567)^8$

Now, $1.515716567^8$:

We know that $a^8=(a^4)^2=(a^2)^4$

$a = 1.515716567$

$a^2=1.515716567\times1.515716567 = 2.297448714$

$a^4=(2.297448714)^2 = 5.278099203$

$a^8=(5.278099203)^2 = 27.85703704$

So, approximately 27.86 when rounded to two decimal places. Wait, but let's check with the exponent rule: $(-8)^{8/5}=e^{\frac{8}{5}\ln(-8)}$, but $\ln(-8)$ is complex, but since we are dealing with real numbers, the principal real root is considered, so $(-8)^{8/5}= (8)^{8/5}\times(-1)^{8/5}$. Since $(-1)^{8/5}=((-1)^{1/5})^8=(-1)^8 = 1$ (because 5th root of -1 is -1, and -1 to the 8th power is 1). So $(-8)^{8/5}=8^{8/5}=8^{1 + 3/5}=8\times8^{3/5}$. $8^{3/5}=(8^{1/5})^3\approx(1.5157)^3\approx3.476$, so $8\times3.476 = 27.808\approx27.81$. Ah! Here's the mistake earlier: $(-1)^{8/5}=((-1)^{1/5})^8=(-1)^8 = 1$, because the 5th root of -1 is -1, and -1 to the 8th power is 1. So $(-8)^{8/5}=8^{8/5}$, because the negative sign raised to an even power (8/5 has numerator 8, which is even) becomes positive. So $8^{8/5}=8^{1.6}$. Let's calculate $8^{1.6}$:

$8^{1}=8$, $8^{0.6}=8^{3/5}=(8^{1/5})^3\approx(1.5157)^3\approx3.476$, so $8\times3.476 = 27.808\approx27.81$. Wait, but when we calculated $1.5157^8$ we got approximately 27.857. There is a difference because $1.5157$ is an approximation of $8^{1/5}$. Let's calculate $8^{1/5}$ more accurately: $8^{1/5}=e^{\ln(8)/5}\approx e^{2.079441542/5}\approx e^{0.415888308}\approx1.515716567$, which is correct. Then $8^{8/5}=(8^{1/5})^8\approx(1.515716567)^8$. Let's compute $(1.515716567)^8$:

$(1.515716567)^2 = 2.297448714$

$(2.297448714)^2 = 5.278099203$

$(5.278099203)^2 = 27.85703704$

Ah, so the correct value is approximately 27.86 when using the more accurate exponentiation. The mistake in the previous exponent rule approach was in the sign: $(-8)^{8/5}=((-8)^{1/5})^8$, and $(-8)^{1/5}$ is a real number (since 5 is odd), so $(-8)^{1/5}=-8^{1/5}$, so $((-8)^{1/5})^8=(-8^{1/5})^8=(8^{1/5})^8=8^{8/5}$, because the negative sign is raised to an even power (8), so it becomes positive. So $8^{8/5}=(8^{1/5})^8\approx(1.515716567)^8\approx27.86$. Wait, but when we calculate $8^{8/5}$ as $8^{1.6}$, let's use the calculator to compute $8^{1.6}$:

Using a calculator, $8^{1.6}=e^{1.6\ln(8)}\approx e^{1.6\times2.079441542}\approx e^{3.327106467}\approx27.857$, which matches the $(1.5157)^8$ calculation. So the correct value is approximately 27.86 when rounded to two decimal places. Wait, but let's check with a calculator:

First, calculate $\sqrt[5]{-8}$: $\sqrt[5]{-8}=-1.515716567$

Then, raise to the 8th power: $(-1.515716567)^8 = (1.515716567)^8\approx27.86$ (using a calculator for the 8th power).

So the correct approximation is approximately 27.86.

Answer:

$\approx\boxed{27.86}$ (or approximately 27.81 if we use a different method, but the more accurate calculation with the calculator gives approximately 27.86) Wait, let's use a calculator directly:

Using a calculator, first compute $\sqrt[5]{-8}$:

$\sqrt[5]{-8} = -8^{1/5} \approx -1.515716567$

Then, raise to the 8th power:

$(-1.515716567)^8 = (1.515716567)^8 \approx 27.857 \approx 27.86$ (rounded to two decimal places).

So the answer is approximately 27.86.